The appropriate choice is ...
.... A. The student failed to keep both sides of the equation balanced
_____
The one rule I learned in my study of Algebra that makes it all make sense is "keep the equal sign sacred." That is, whatever you do to one side of the equation must also be done to the other side of the equation. (Various teachers show various shortcuts that make it look like you're doing different things to the different sides of the equation. Don't be confused. If the operation can't be put in terms of doing the same thing to both sides, it doesn't work.)
Of course, you can always add 0 or multiply by 1 whenever and wherever you want.
_____
For the above equation, the appropriate solution is
... a + 3 = 9
... a + 3 - 3 = 9 - 3 . . . . . . addition property of equality
... a + 0 = 6 . . . . . . . . . . . .rules of arithmetic
... a = 6 . . . . . . . . . . . . . . . additive identity property of 0
